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Theorem subccocl 13735
Description: A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcidcl.2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subcidcl.x  |-  ( ph  ->  X  e.  S )
subccocl.o  |-  .x.  =  (comp `  C )
subccocl.y  |-  ( ph  ->  Y  e.  S )
subccocl.z  |-  ( ph  ->  Z  e.  S )
subccocl.f  |-  ( ph  ->  F  e.  ( X J Y ) )
subccocl.g  |-  ( ph  ->  G  e.  ( Y J Z ) )
Assertion
Ref Expression
subccocl  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) )

Proof of Theorem subccocl
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcidcl.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
2 eqid 2296 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
3 eqid 2296 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
4 subccocl.o . . . . 5  |-  .x.  =  (comp `  C )
5 subcrcl 13709 . . . . . 6  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
61, 5syl 15 . . . . 5  |-  ( ph  ->  C  e.  Cat )
7 subcidcl.2 . . . . 5  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
82, 3, 4, 6, 7issubc2 13729 . . . 4  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  (  Homf 
`  C )  /\  A. x  e.  S  ( ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >.  .x.  z
) f )  e.  ( x J z ) ) ) ) )
91, 8mpbid 201 . . 3  |-  ( ph  ->  ( J  C_cat  (  Homf  `  C )  /\  A. x  e.  S  (
( ( Id `  C ) `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >.  .x.  z
) f )  e.  ( x J z ) ) ) )
109simprd 449 . 2  |-  ( ph  ->  A. x  e.  S  ( ( ( Id
`  C ) `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) )
11 subcidcl.x . . 3  |-  ( ph  ->  X  e.  S )
12 subccocl.y . . . . . 6  |-  ( ph  ->  Y  e.  S )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  S )
14 subccocl.z . . . . . . 7  |-  ( ph  ->  Z  e.  S )
1514ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  S )
16 subccocl.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( X J Y ) )
1716ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( X J Y ) )
18 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
19 simplr 731 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
2018, 19oveq12d 5892 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x J y )  =  ( X J Y ) )
2117, 20eleqtrrd 2373 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( x J y ) )
22 subccocl.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( Y J Z ) )
2322ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( Y J Z ) )
24 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  y  =  Y )
25 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  z  =  Z )
2624, 25oveq12d 5892 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  (
y J z )  =  ( Y J Z ) )
2723, 26eleqtrrd 2373 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( y J z ) )
28 simp-5r 745 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  x  =  X )
29 simp-4r 743 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  y  =  Y )
3028, 29opeq12d 3820 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  <. x ,  y >.  =  <. X ,  Y >. )
31 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  z  =  Z )
3230, 31oveq12d 5892 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
33 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  g  =  G )
34 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  f  =  F )
3532, 33, 34oveq123d 5895 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
g ( <. x ,  y >.  .x.  z
) f )  =  ( G ( <. X ,  Y >.  .x. 
Z ) F ) )
3628, 31oveq12d 5892 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
x J z )  =  ( X J Z ) )
3735, 36eleq12d 2364 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  <->  ( G
( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
3827, 37rspcdv 2900 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  ( A. g  e.  (
y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
3921, 38rspcimdv 2898 . . . . . 6  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  ( A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4015, 39rspcimdv 2898 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( A. z  e.  S  A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4113, 40rspcimdv 2898 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  S  A. z  e.  S  A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4241adantld 453 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( ( ( Id
`  C ) `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) )  -> 
( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) ) )
4311, 42rspcimdv 2898 . 2  |-  ( ph  ->  ( A. x  e.  S  ( ( ( Id `  C ) `
 x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) )  -> 
( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) ) )
4410, 43mpd 14 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874  compcco 13236   Catccat 13582   Idccid 13583    Homf chomf 13584    C_cat cssc 13700  Subcatcsubc 13702
This theorem is referenced by:  subccatid  13736  funcres  13786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-ixp 6834  df-ssc 13703  df-subc 13705
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